Question 1

Let $f(x) = \sqrt{x}$ and $g(x) = x^2 + 1$. What is the formula for the composite function $(f \circ g)(x)$?

Accepted Answer

$\sqrt{x^2 + 1}$

Question 2

Given $f(x) = \frac{1}{x-2}$ and $g(x) = \sqrt{x}$, what is the domain of $(f+g)(x)$?

Accepted Answer

$[0,2) \cup (2, \infty)$

Question 3

If $f(x) = x^2$ and $g(x) = x-3$, what is $(g \circ f)(7)$?

Accepted Answer

$4$

Question 4

Suppose $f(x) = \sqrt{x}$ and $g(x) = x^2 + 1$. Which of the following best describes the process to compute $(f \circ g)(x)$?

Accepted Answer

First square $x$, add 1, then take the square root

Question 5

Given $f(x) = \frac{1}{x-2}$ and $g(x) = \sqrt{x}$, what is the domain of $\left(\frac{f}{g}\right)(x)$?

Accepted Answer

$(0,2) \cup (2, \infty)$

Question 6

If $f(x) = x^2$ and $g(x) = x-3$, what is the formula for $(f \circ g)(x)$?

Accepted Answer

$(x-3)^2$

Question 7

A balloon's radius $R$ is a function of the volume $V$ of air pumped in, and $V$ is a function of time $t$. Which expression gives the radius as a function of time?

Accepted Answer

$R(V(t))$

Question 8

Let $f(x) = \sqrt{x}$ and $g(x) = x^2 + 1$. What is the domain of $(f \circ g)(x)$?

Accepted Answer

$(-\infty, \infty)$

Question 9

Suppose $f(x) = \sqrt{x+9}$. Find functions $g$ and $h$ such that $f = g \circ h$.

Accepted Answer

$g(x) = \sqrt{x},\ h(x) = x+9$

Question 10

A ship sails at 20 mi/h parallel to a straight shoreline and passes a lighthouse at noon. The distance $s$ from the lighthouse after $t$ hours is given by $s = \sqrt{25 + (20t)^2}$. What does this function represent?

Accepted Answer

The distance from the lighthouse to the ship as a function of time

Question 11

If $f(x) = x^2$ and $g(x) = x+3$, what is $(f \circ g \circ h)(x)$ where $h(x) = x-1$?

Accepted Answer

$((x-1)+3)^2$

Question 12

Given $f(x) = \frac{1}{x-2}$ and $g(x) = \sqrt{x}$, what is $(f \cdot g)(x)$?

Accepted Answer

$\frac{\sqrt{x}}{x-2}$

Question 13

Which of the following is true about the composition of functions $f$ and $g$?

Accepted Answer

$(f \circ g)(x)$ means apply $g$ first, then $f$

Question 14

If $f(x) = x^2$ and $g(x) = x-3$, what is $(f \circ g)(5)$?

Accepted Answer

$4$

Question 15

Suppose $f(x) = \sqrt{x}$ and $g(x) = x^2 + 1$. Which of the following is the correct decomposition of $h(x) = \sqrt{x^2 + 1}$?

Accepted Answer

$h(x) = f(g(x))$

Question 16

If $f(x) = \sqrt{x}$ and $g(x) = x^2 + 1$, what is the value of $(f \circ g)(2)$?

Accepted Answer

$\sqrt{5}$

Question 17

Given $f(x) = \frac{1}{x-2}$ and $g(x) = \sqrt{x}$, what is $(f-g)(x)$?

Accepted Answer

$\frac{1}{x-2} - \sqrt{x}$

Question 18

Which of the following is a real-world example of function composition?

Accepted Answer

Calculating the distance from a lighthouse to a ship as a function of time, where distance depends on the ship's position, which depends on time

Combining Functions: Sums, Products, Quotients, and Composition

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